Closed loop feedback in mimo systems

ABSTRACT

Feedback bandwidth may be reduced in a closed loop MIMO system by factoring non essential information out of a beamforming matrix.

RELATED APPLICATION

This application is a Continuation of U.S. Nonprovisional applicationSer. No. 10/939,130, by Lin et al., filed Sep. 10, 2004, which isincorporated herein by reference in its entirety for all purposes.

FIELD

The present invention relates generally to wireless networks, and morespecifically to wireless networks that utilize multiple spatialchannels.

BACKGROUND

Closed loop multiple-input-multiple-output (MIMO) systems typicallytransmit channel state information from a receiver to a transmitter. Thetransmitter may then utilize the information to do beam forming.Transmitting the channel state information consumes bandwidth that mightotherwise be available for data traffic.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a diagram of two wireless stations;

FIGS. 2 and 3 show flowcharts in accordance with various embodiments ofthe present invention;

FIG. 4 shows an electronic system in accordance with various embodimentsof the present invention; and

FIGS. 5 and 6 show flowcharts in accordance with various embodiments ofthe present invention.

DESCRIPTION OF EMBODIMENTS

In the following detailed description, reference is made to theaccompanying drawings that show, by way of illustration, specificembodiments in which the invention may be practiced. These embodimentsare described in sufficient detail to enable those skilled in the art topractice the invention. It is to be understood that the variousembodiments of the invention, although different, are not necessarilymutually exclusive. For example, a particular feature, structure, orcharacteristic described herein in connection with one embodiment may beimplemented within other embodiments without departing from the spiritand scope of the invention. In addition, it is to be understood that thelocation or arrangement of individual elements within each disclosedembodiment may be modified without departing from the spirit and scopeof the invention. The following detailed description is, therefore, notto be taken in a limiting sense, and the scope of the present inventionis defined only by the appended claims, appropriately interpreted, alongwith the full range of equivalents to which the claims are entitled. Inthe drawings, like numerals refer to the same or similar functionalitythroughout the several views.

FIG. 1 shows a diagram of two wireless stations: station 102, andstation 104. In some embodiments, stations 102 and 104 are part of awireless local area network (WLAN). For example, one or more of stations102 and 104 may be an access point in a WLAN. Also for example, one ormore of stations 102 and 104 may be a mobile station such as a laptopcomputer, personal digital assistant (PDA), or the like. Further, insome embodiments, stations 102 and 104 are part of a wireless wide areanetwork (WWAN).

In some embodiments, stations 102 and 104 may operate partially incompliance with, or completely in compliance with, a wireless networkstandard. For example, stations 102 and 104 may operate partially incompliance with a standard such as ANSI/IEEE Std. 802.11, 1999 Edition,although this is not a limitation of the present invention. As usedherein, the term “802.11” refers to any past, present, or future IEEE802.11 standard, including, but not limited to, the 1999 edition. Alsofor example, stations 102 and 104 may operate partially in compliancewith any other standard, such as any future IEEE personal area networkstandard or wide area network standard.

Stations 102 and 104 each include multiple antennas. Each of stations102 and 104 includes “N” antennas, where N may be any number. In someembodiments, stations 102 and 104 have an unequal number of antennas.The remainder of this description discusses the case where stations 102and 104 have an equal number of antennas, but the various embodiments ofthe invention are not so limited. The “channel” through which stations102 and 104 communicate may include many possible signal paths. Forexample, when stations 102 and 104 are in an environment with many“reflectors” (e.g. walls, doors, or other obstructions), many signalsmay arrive from different paths. This condition is known as “multipath.”In some embodiments, stations 102 and 104 utilize multiple antennas totake advantage of the multipath and to increase the communicationsbandwidth. For example, in some embodiments, stations 102 and 104 maycommunicate using Multiple-Input-Multiple-Output (MIMO) techniques. Ingeneral, MIMO systems offer higher capacities by utilizing multiplespatial channels made possible by multipath.

In some embodiments, stations 102 and 104 may communicate usingorthogonal frequency division multiplexing (OFDM) in each spatialchannel. Multipath may introduce frequency selective fading which maycause impairments like inter-symbol interference (ISI). OFDM iseffective at combating frequency selective fading in part because OFDMbreaks each spatial channel into small subchannels such that eachsubchannel exhibits a more flat channel characteristic. Scalingappropriate for each subchannel may be implemented to correct anyattenuation caused by the subchannel. Further, the data carryingcapacity of each subchannel may be controlled dynamically depending onthe fading characteristics of the subchannel.

MIMO systems may operate either “open loop” or “closed loop.” In openloop MIMO systems, a station estimates the state of the channel withoutreceiving channel state information directly from another station. Ingeneral, open loop systems employ exponential decoding complexity toestimate the channel. In closed loop systems, communications bandwidthis utilized to transmit current channel state information betweenstations, thereby reducing the necessary decoding complexity, and alsoreducing overall throughput. The communications bandwidth used for thispurpose is referred to herein as “feedback bandwidth.” When feedbackbandwidth is reduced in closed loop MIMO systems, more bandwidth isavailable for data communications.

The current channel state information may be represented by an N×Nunitary beamforming matrix V determined using a singular valuedecomposition (SVD) algorithm, and the transmitter may process anoutgoing signal using the beamforming matrix V to transmit into multiplespatial channels. In a straightforward implementation, the receiversends each element of the unitary matrix V back to transmitter. Thisscheme involves sending information related to the 2N² real numbers forany N×N complex unitary matrix, where N is the number of spatialchannels in MIMO system.

In some embodiments of the present invention, the beamforming matrix Vis represented by N²−N real numbers instead of 2N² real numbers. Bysending N²−N real numbers instead of 2N² real numbers to represent thebeamforming matrix, the feedback bandwidth may be reduced. Non-essentialinformation may be factored out of the beamforming matrix and discardedprior to quantizing parameters that are used to represent thebeamforming matrix. For example, non-essential phase information may befactored from each column in the beamforming matrix, and then N²−Nparameters may be utilized to represent the matrix without thenon-essential phase information.

A mathematical background of the SVD operation is provided below, andthen examples are provided for 2×2 and 3×3 MIMO systems. In the 2×2closed loop MIMO example, two angles in [0, π/2] and (π, −π] are used asfeedback parameters. Compared to the straightforward example above, thevarious embodiments of the present invention represented by the 2×2example below reduce the amount of feedback from eight real numbers totwo real numbers per subcarrier. In the 3×3 closed loop MIMO example,one sign bit plus four angles between [0, π/2] and two angles between[−π, π] are used as feedback parameters. Compared to the straightforwardexample above, the various embodiments of the present inventionrepresented by the 3×3 example below reduce the amount of feedback from18 real numbers to six real numbers per subcarrier.

A transmit beamforming matrix may be found using SVD as follows:

H=UDV′  (1)

x=Vd  (2)

where d is the N-vector of code bits for N data streams; x is thetransmitted signal vector on the antennas; H is the channel matrix; H'ssingular value decomposition is H=UDV′; U and V are unitary; D is adiagonal matrix with H's eigenvalues; V is N×N, and N is the number ofspatial channels. To obtain V at the transmitter, the transmitter maysend training symbols to the receiver; the receiver may compute thematrix V′; and the receiver may feedback parameters representing V tothe transmitter. As described more fully below, the number of feedbackparameters used to represent V may be reduced by factoring non-essentialphase information from V′ and discarding it prior to quantizing theparameters.

2×2 Beamforming Matrices

Any complex 2×2 matrix may be written as

$\begin{matrix}{V = {\begin{pmatrix}{b_{11}^{{\varphi}_{11}}} & {b_{12}^{{\varphi}_{12}}} \\{b_{21}^{{\varphi}_{21}}} & {b_{22}^{{\varphi}_{22}}}\end{pmatrix}.}} & (3)\end{matrix}$

If V is unitary i.e., VV′=I, then

$\begin{matrix}{V = \begin{pmatrix}{b_{11}^{{\varphi}_{11}}} & {b_{12}^{{\varphi}_{12}}} \\{{- b_{12}}^{{\varphi}_{21}}} & {b_{11}^{{({\varphi_{12} + \varphi_{21} - \varphi_{11}})}}}\end{pmatrix}} & (4)\end{matrix}$

where b₁₁ ²+b₁₂ ²=1. We can further limit b₁₁ε[0,1], b₁₂ε[0,1],φ_(ij)ε[−π, π) without loss of generality. There are 4 degrees offreedom in V. After factoring the common phases for each row and column,the unitary matrix V can be written as

$\begin{matrix}{V = {{\begin{pmatrix}1 & 0 \\0 & ^{{({\varphi_{21} - \varphi_{11}})}}\end{pmatrix}\begin{pmatrix}b_{11} & b_{12} \\{- b_{12}} & b_{11}\end{pmatrix}\begin{pmatrix}^{{\varphi}_{11}} & 0 \\0 & ^{{\varphi}_{12}}\end{pmatrix}} = {P_{L}\overset{\sim}{V}P_{R}}}} & (5)\end{matrix}$

where P_(L) and P_(R) are pure phase matrices and diagonal. (510, FIG.5) P_(R) is generated by factoring phase values from each column of V,and P_(L) is found by factoring phase values from each row of V. {tildeover (V)} is a magnitude matrix that has entries consisting of scalarquantities that represent the magnitudes of the entries of V. Since b₁₁²+b₁₂ ²=1, {tilde over (V)} can be written as

$\begin{matrix}{{\overset{\sim}{V} = \begin{pmatrix}{\cos \; \theta} & {\sin \; \theta} \\{{- \sin}\; \theta} & {\cos \; \theta}\end{pmatrix}},{{{where}\mspace{14mu} \theta} \in {\left\lbrack {0,\frac{\pi}{2}} \right\rbrack.}}} & (6)\end{matrix}$

In various embodiments of the present invention, only two angles i.e., θand φ₁₁-φ₂₁ are fed back to the transmitter. (530, FIG. 5) The firstangle, θ, unambiguously represents {tilde over (V)}, and the secondangle, φ₁₁-φ₂₁, unambiguously represents P_(L). (520, FIG. 5) In otherembodiments of the present invention, a trigonometric function of θ maybe selected as a parameter to feed back. For example, cos θ may be fedback as a parameter to represent {tilde over (V)}. In still furtherembodiments, another parameter may be selected that may unambiguouslydescribe {tilde over (V)}.

The phase information in P_(R) may be discarded. Equation (1) can berewritten as

$\begin{matrix}\begin{matrix}{H = {U\; D\; V^{\prime}}} \\{= {U\; {D\left( {P_{L}\overset{\sim}{V}P_{R}} \right)}^{\prime}}} \\{= {U\; \underset{\underset{\overset{\sim}{D}}{}}{D\; P_{R}^{\prime}}\left( \underset{\underset{\overset{\_}{V}}{}}{P_{L}\overset{\sim}{V}} \right)^{\prime}}} \\{= {\underset{\overset{\sim}{U}}{\underset{}{U\; P_{R}^{\prime}}}{D\left( \underset{\underset{\overset{\_}{V}}{}}{P_{L}\overset{\sim}{V}} \right)}^{\prime}}}\end{matrix} & (7)\end{matrix}$

where we have used the fact that D and P′_(R) are diagonal and thereforecommute. It should be noted that H=ŨD V′ is also a singular valuedecomposition of H. For the SVD algorithm, the change from U to Ũ onlychanges the multiplication matrix on the receiver side. When H is a m×nmatrix with m≠n, we can still write H=U{tilde over (D)} V and the effectof beam forming with V amounts to a rotation in the I/Q plane, which maybe taken care of by the training process. Therefore, feeding back V tothe transmitter is sufficient for the SVD algorithm. Since V is fullydetermined by θ and φ₁₁-φ₂₁, only two angles are required to feedbackand they are between

$\left\lbrack {0,\frac{\pi}{2}} \right\rbrack \mspace{14mu} {and}\mspace{14mu} {\left( {{- \pi},\pi} \right\rbrack.}$

As stated above, the unitary matrix V may be factored into the productof three matrices:

$\begin{matrix}\begin{matrix}{V = {\begin{pmatrix}1 & 0 \\0 & ^{{({\varphi_{21} - \varphi_{11}})}}\end{pmatrix}\begin{pmatrix}b_{11} & b_{12} \\{- b_{12}} & b_{11}\end{pmatrix}\begin{pmatrix}^{{\varphi}_{11}} & 0 \\0 & ^{{\varphi}_{12}}\end{pmatrix}}} \\{= {\begin{pmatrix}1 & 0 \\0 & ^{{({\varphi_{21} - \varphi_{11}})}}\end{pmatrix}\begin{pmatrix}{\cos \; \theta} & {\sin \; \theta} \\{{- \sin}\; \theta} & {\cos \; \theta}\end{pmatrix}\begin{pmatrix}^{{\varphi}_{11}} & 0 \\0 & ^{{\varphi}_{12}}\end{pmatrix}}}\end{matrix} & (8)\end{matrix}$

where θ and φ₂₁-φ₁₁ are between

$\left\lbrack {0,\frac{\pi}{2}} \right\rbrack \mspace{14mu} {and}\mspace{14mu} {\left( {{- \pi},\pi} \right\rbrack.}$

The parameters θ and φ₂₁-φ₁₁ may be obtained at the receiver as follows:

$\begin{matrix}{{\theta = {\arccos \left( {{abs}\left( v_{11} \right)} \right)}},{\theta \in \left\lbrack {0,{\pi/2}} \right\rbrack}} & (9) \\{\varphi_{ij} = \left\{ \begin{matrix}{{{\arctan \left( \frac{{Im}\left( v_{ij} \right)}{{Re}\left( v_{ij} \right)} \right)} + {\pi/2}},} & {{{Im}\left( v_{ij} \right)} \geq 0} \\{{{\arctan \left( \frac{{Im}\left( v_{ij} \right)}{{Re}\left( v_{ij} \right)} \right)} + {3{\pi/2}}},} & {{{Im}\left( v_{ij} \right)} < 0}\end{matrix} \right.} & (10)\end{matrix}$

and the receiver may quantize θ and φ₂₁-φ₁₁ and feed them back to thetransmitter as parameters that represent V. The transmitter mayreconstruct V by determining the amplitudes using θ, and applying aphase rotation to the bottom row using φ₂₁-φ₁₁.

$\begin{matrix}{\overset{\_}{V} = \begin{pmatrix}{\cos \; \theta} & {\sin \; \theta} \\{{- \sin}\; {\theta }^{{({\varphi_{21} - \varphi_{11}})}}} & {\cos \; {\theta }^{{({\varphi_{21} - \varphi_{11}})}}}\end{pmatrix}} & (11)\end{matrix}$

The transmitter may then use V for beamforming:

x= Vd  (12)

3×3 Beamforming Matrices

Any complex, unit 3-vector may be written as

$\begin{matrix}{{v = {\begin{bmatrix}v_{1} \\v_{2} \\v_{3}\end{bmatrix} = {^{{\theta}_{1}}\begin{bmatrix}{\cos \left( \varphi_{1} \right)} \\{{\sin \left( \varphi_{1} \right)}{\cos \left( \varphi_{2} \right)}^{{\theta}_{2}}} \\{{\sin \left( \varphi_{1} \right)}{\sin \left( \varphi_{2} \right)}^{{\theta}_{3}}}\end{bmatrix}}}}{{{{where}\mspace{14mu} {v}^{2}} = {{{v_{1}}^{2} + {v_{2}}^{2} + {v_{3}}^{2}} = 1}};}{\varphi_{1},{\varphi_{2} \in {\left\lfloor {0,{\pi/2}} \right\rfloor \mspace{14mu} {and}\mspace{14mu} \theta_{1}}},\theta_{2},{\theta_{3} \in {\left\lbrack {{- \pi},\pi} \right).}}}} & (13)\end{matrix}$

Further, any unitary 3 by 3 matrix may be written as

$\begin{matrix}{{{V = {\left\lbrack {v_{1}\mspace{14mu} v_{2}\mspace{14mu} v_{3}} \right\rbrack =}}\quad}{\quad\left\lbrack \begin{matrix}{^{{\theta}_{11}}{\cos \left( \varphi_{11} \right)}} & {^{{\theta}_{12}}{\cos \left( \varphi_{12} \right)}} & {^{{\theta}_{13}}{\cos \left( \varphi_{13} \right)}} \\{^{{\theta}_{11}}^{{\theta}_{21}}{\sin \left( \varphi_{11} \right)}{\cos \left( \varphi_{21} \right)}} & {^{{\theta}_{12}}^{{\theta}_{22}}{\sin \left( \varphi_{12} \right)}{\cos \left( \varphi_{22} \right)}} & {^{{\theta}_{13}}^{{\theta}_{23}}{\sin \left( \varphi_{13} \right)}{\cos \left( \varphi_{23} \right)}} \\{^{{\theta}_{11}}^{{\theta}_{31}}{\sin \left( \varphi_{11} \right)}{\sin \left( \varphi_{21} \right)}} & {^{{\theta}_{12}}^{{\theta}_{32}}{\sin \left( \varphi_{12} \right)}{\sin \left( \varphi_{22} \right)}} & {^{{\theta}_{13}}^{{\theta}_{33}}{\sin \left( \varphi_{13} \right)}{\sin \left( \varphi_{23} \right)}}\end{matrix} \right\rbrack}} & (14)\end{matrix}$

where v′_(j)v_(j)=1 and v′_(j)v_(k)=0 for j,k=1, 2, 3. The phases on thefirst row and the first column can be factored as the product of thefollowing three matrices:

$\begin{matrix}{V = \underset{\underset{P_{L}}{}}{\begin{bmatrix}1 & 0 & 0 \\0 & ^{{\theta}_{21}} & 0 \\0 & 0 & ^{{\theta}_{31}}\end{bmatrix}}} & (15) \\\underset{\overset{\sim}{V}}{\underset{}{\begin{bmatrix}{\cos \left( \varphi_{11} \right)} & {\cos \left( \varphi_{12} \right)} & {\cos \left( \varphi_{13} \right)} \\{{\sin \left( \varphi_{11} \right)}{\cos \left( \varphi_{21} \right)}} & {^{{\phi}_{22}}{\sin \left( \varphi_{12} \right)}{\cos \left( \varphi_{22} \right)}} & {^{{\phi}_{23}}{\sin \left( \varphi_{13} \right)}{\cos \left( \varphi_{23} \right)}} \\{{\sin \left( \varphi_{11} \right)}{\sin \left( \varphi_{21} \right)}} & {^{{\phi}_{32}}{\sin \left( \varphi_{12} \right)}{\sin \left( \varphi_{22} \right)}} & {^{{\phi}_{33}}{\sin \left( \varphi_{13} \right)}{\sin \left( \varphi_{23} \right)}}\end{bmatrix}}} & \; \\\underset{\underset{P_{R}}{}}{\begin{bmatrix}^{{\theta}_{11}} & 0 & 0 \\0 & ^{{\theta}_{12}} & 0 \\0 & 0 & ^{{\theta}_{13}}\end{bmatrix}} & \;\end{matrix}$

where P₁ and P_(R) are pure phase matrices and diagonal. P_(R) isgenerated by factoring phase values from each column of V, and P_(L) isfound by factoring phase values from each row of V, and whereφ_(jk)ε└0,π/2┘ and cos(φ_(jk)), cos(φ_(jk)), sin(φ_(jk))≧0. {tilde over(V)} is a magnitude matrix that includes all of the magnitudeinformation originally present in the entries of V. As used herein, theterm “magnitude matrix” refers to a matrix that remains after P₁ andP_(R) are factored out of the original beamforming matrix. As shown inthe above example, one or more entries in a magnitude matrix may includephase information. It should be noted that {tilde over (V)}=[{tilde over(v)}₁ {tilde over (v)}₂ {tilde over (v)}₃] is still unitary since thephase factorization doesn't change the unitary property.

In various embodiments of the present invention, two parameters arechosen to represent P_(L), four parameters are chosen to represent{tilde over (V)}, and P_(R) is discarded. In some embodiments, theangles θ₂₁, θ₃₁ are selected as parameters to represent P_(L). Matrix{tilde over (V)} can be determined by four parameters and a sign bit,and there are many combinations of the four parameters that are subsetsof all the angles in {tilde over (V)}. Different combinations result indifferent complexities in the reconstruction of {tilde over (V)} at thetransmitter. It should be noted that the complexity of extracting allthe angles of {tilde over (V)} is relatively low compared to that of theconstruction of {tilde over (V)} based on four parameters. Instead ofdirectly sending angles back, some embodiments may send functions of theselected four angles back. For example, common trigonometric functionssuch as sin( ), cos( ), and tan( ) may be selected. The variousembodiments of the present invention contemplate all possible sets offour parameters to represent {tilde over (V)}. One set of fourparameters φ₁₁, φ₁₂, φ₂₁, φ₂₂ and the sign of φ₂₂ provide a solutionthat is now elaborated. The extraction of the angles φ₁₁, φ₁₂, φ₂₁, φ₂₂may be performed as:

φ₁₁=arccos(|v₁₁|)  (16)

φ₁₂=arccos(|v₁₂|)  (17)

φ₁₂=arctan(|v₃₁|/|v₂₁|)  (18)

φ₂₂=arctan(|v₃₂|/|v₂₂|)  (19)

It should be noted that φ₁₁, φ₁₂, φ₂₁, φ₂₂ are all within [0,π/2]instead of [0,π] and the sign of φ₂₂ takes only one bit. In variousembodiments, the feedback includes one angle in [0,π] and three anglesin [0,π/2].

In embodiments using the above parameters to represent P_(L) and {tildeover (V)}, the receiver quantizes θ₂₁, θ₃₁, φ₁₁, φ₁₂, φ₂₁, φ₂₂ and feedsthem back to the transmitter along with sign(φ₂₂), which can be found assign(φ₂₂)=sign(angle({tilde over (v)}₂₂)). (620, 630, FIG. 6)

The receiver may receive the parameters, reconstruct {tilde over (V)},and perform beamforming. The outline of the reconstruction of {tildeover (V)} is now shown as: computation of φ₂₂, φ₃₂ to reconstruct {tildeover (v)}₂, the second column of {tilde over (V)}; and computation of{tilde over (v)}₃, the third column of {tilde over (V)} using theunitary property of {tilde over (V)}. We rewrite {tilde over (V)} as

$\begin{matrix}{\overset{\sim}{V} = \begin{bmatrix}{\cos \left( \varphi_{11} \right)} & {\cos \left( \varphi_{12} \right)} & {\overset{\sim}{v}}_{13} \\{{\sin \left( \varphi_{11} \right)}{\cos \left( \varphi_{21} \right)}} & {^{{\phi}_{22}}{\sin \left( \varphi_{12} \right)}{\cos \left( \varphi_{22} \right)}} & {\overset{\sim}{v}}_{23} \\{{\sin \left( \varphi_{11} \right)}{\sin \left( \varphi_{21} \right)}} & {^{{\phi}_{32}}{\sin \left( \varphi_{12} \right)}{\sin \left( \varphi_{22} \right)}} & {\overset{\sim}{v}}_{33}\end{bmatrix}} & (20)\end{matrix}$

Since {tilde over (v)}₂ is orthogonal to {tilde over (v)}₁, we havev′₁v₂=0 or

c ₁ +c ₂ e ^(iφ) ²² +c ₂ e ^(iφ) ³² =0  (21)

where

c ₁=cos(φ₁₁)cos(φ₁₂)

c ₂=sin(φ₁₁)cos(φ₂₁)sin(φ₁₂)cos(φ₂₂)

c ₃=sin(φ₁₁)sin(φ₂₁)sin(φ₁₂)sin(φ₂₂)  (22)

The c_(j) are all greater than or equal to zero since φ₁₁, φ₁₂, φ₂₁, φ₂₂are all within [0,π/2]. Equation (21) can be explicitly solved by usinglaws of cosine. The solutions of φ₂₂, φ₃₂ are

$\begin{matrix}{{\phi_{22} = {{{sign}\left( \phi_{22} \right)}\left\lbrack {\arccos\left( \frac{c_{1}^{2} + c_{2}^{2} - c_{3}^{2}}{2c_{1}c_{2}} \right)} \right\rbrack}}{\phi_{32} = {- {{{sign}\left( \phi_{22} \right)}\left\lbrack {\arccos\left( \frac{c_{1}^{2} + c_{3}^{2} - c_{2}^{2}}{2c_{1}c_{3}} \right)} \right\rbrack}}}} & (23)\end{matrix}$

Since {tilde over (V)}′ is also unitary, the norm of the first row is 1.Considering {tilde over (v)}₁₃=cos(φ₁₃) is a positive number, we solve{tilde over (v)}₁₃ as

{tilde over (v)} ₁₃=√{square root over (1−cos²(φ₁₁)−cos²(φ₁₂))}{squareroot over (1−cos²(φ₁₁)−cos²(φ₁₂))}  (24)

Since {tilde over (V)}′ is unitary, the second row of {tilde over (V)}is orthogonal to the second row. {tilde over (v)}₂₃ can be solved as

$\begin{matrix}{{\overset{\sim}{v}}_{23} = \frac{\begin{matrix}{{{- {\cos \left( \varphi_{11} \right)}}{\sin \left( \varphi_{11} \right)}{\cos \left( \varphi_{21} \right)}} -} \\{\cos \left( \varphi_{12} \right){\sin \left( \varphi_{12} \right)}{\cos \left( \varphi_{22} \right)}^{{\phi}_{22}}}\end{matrix}}{\sqrt{1 - {\cos^{2}\left( \varphi_{11} \right)} - {\cos^{2}\left( \varphi_{12} \right)}}}} & (25)\end{matrix}$

Similarly, {tilde over (v)}₃₃ is

$\begin{matrix}{{\overset{\sim}{v}}_{33} = \frac{\begin{matrix}{{{- {\cos \left( \varphi_{11} \right)}}{\sin \left( \varphi_{11} \right)}{\sin \left( \varphi_{21} \right)}} -} \\{{\cos \left( \varphi_{12} \right)}{\sin \left( \varphi_{12} \right)}{\sin \left( \varphi_{22} \right)}^{{\phi}_{32}}}\end{matrix}}{\sqrt{1 - {\cos^{2}\left( \varphi_{11} \right)} - {\cos^{2}\left( \varphi_{12} \right)}}}} & (26)\end{matrix}$

Remembering that

$\begin{matrix}{{P_{L} = \begin{bmatrix}1 & 0 & 0 \\0 & ^{{\theta}_{21}} & 0 \\0 & 0 & ^{{\theta}_{31}}\end{bmatrix}},} & (27)\end{matrix}$

beamforming may be performed as:

x=P_(L){tilde over (V)}d  (28)

FIG. 2 shows a flowchart in accordance with various embodiments of thepresent invention. In some embodiments, method 200 may be used in, orfor, a wireless system that utilizes MIMO technology. In someembodiments, method 200, or portions thereof, is performed by a wirelesscommunications device, embodiments of which are shown in the variousfigures. In other embodiments, method 200 is performed by a processor orelectronic system. Method 200 is not limited by the particular type ofapparatus or software element performing the method. The various actionsin method 200 may be performed in the order presented, or may beperformed in a different order. Further, in some embodiments, someactions listed in FIG. 2 are omitted from method 200.

Method 200 is shown beginning at block 210 in which channel stateinformation is estimated from received signals. The channel stateinformation may include the channel state matrix H described above. At220, a beamforming matrix is determined from the channel stateinformation. In some embodiments, this corresponds to performingsingular value decomposition (SVD) as described above with reference toequations (1) and (7). The beamforming matrix V is also described above.

At 230, a phase angle is factored out of each column of the beamformingmatrix. For example, as shown above in equations (5), (8), and (15), thephase matrix P_(R) may be factored out of the beamforming matrix anddiscarded. At 240, additional phase information is factored from thebeamforming matrix to yield a phase matrix and an magnitude matrix. Inthe various embodiments of the present invention described above, theadditional phase information is represented by the phase matrix P_(L),and the magnitude matrix is represented by {tilde over (V)}. Themagnitude matrix includes the magnitude information from the originalbeamforming matrix V, and may or may not include phase information.Accordingly, the entries in {tilde over (V)} may be scalars or complexnumbers.

At 250, the phase matrix and magnitude matrix are represented using N²−Nparameters, where N is a number of spatial channels. For example, in the2×2 embodiments described above, N=2, and the phase matrix and magnitudematrix are represented by two parameters. One parameter, θ, is used torepresent the magnitude matrix and one parameter, φ₁₁-φ₂₁, is used torepresent the phase matrix. Also for example, in the 3×3 embodimentsdescribed above, N=3, and the phase matrix and magnitude matrix arerepresented by six parameters and a sign bit. The phase matrix isrepresented by two parameters, and the magnitude matrix is representedby four parameters and a sign bit. The choice of parameters to representthe magnitude matrix is large.

At 260, the parameters are quantized. They can be quantized individuallyor jointly. The parameters are quantized in the ranges appropriate forthe range of the parameters selected. For example, in the 2×2embodiments described above, θ and φ₁₁-φ₂₁, are quantized between

${\left\lbrack {0,\frac{\pi}{2}} \right\rbrack \mspace{14mu} {and}\mspace{14mu} \left( {{- \pi},\pi} \right\rbrack},$

respectively. At 270, the quantized parameters are transmitted. Thequantized parameters may be transmitted using any type of protocol orany type of communications link, including a wireless link such as awireless link between stations like those described with reference toFIG. 1.

FIG. 3 shows a flowchart in accordance with various embodiments of thepresent invention. In some embodiments, method 300 may be used in, orfor, a wireless system that utilizes MIMO technology. In someembodiments, method 300, or portions thereof, is performed by a wirelesscommunications device, embodiments of which are shown in the variousfigures. In other embodiments, method 300 is performed by a processor orelectronic system. Method 300 is not limited by the particular type ofapparatus or software element performing the method. The various actionsin method 300 may be performed in the order presented, or may beperformed in a different order. Further, in some embodiments, someactions listed in FIG. 3 are omitted from method 300.

Method 300 is shown beginning at block 310 in which at least one angleparameter is received. This may correspond to a transmitter receivingone or more angle parameters that represent a magnitude matrix. Forexample, the at least one angle parameter may include θ as describedabove with reference to equation (6), or may include φ₁₁φ₁₂, φ₂₁, φ₂₂,as described above with reference to equations (15)-(19).

At 320, magnitudes of entries in a beamforming matrix are determinedfrom the at least one angle parameter. For example, as shown in equation(11), the magnitude of the entries in a 2×2 beamforming matrix may bedetermined from the angle parameter θ, and as shown in equations (20)and (24)-(26), the magnitude of the entries in a 3×3 beamforming matrixmay be determined from the angle parameters φ₁₁, φ₁₂, φ₂₁, and φ₂₂.

At 330, at least one phase parameter is received. This may correspond tothe transmitter receiving one or more phase parameters that represent aphase matrix. For example, the at least one phase parameter may includeφ₂₁-φ₁₁ as described above with reference to equations (5) and (8), ormay include φ₁₁, φ₁₂, φ₂₁, φ₂₂, as described above with reference toequations (15)-(19). At 340, the at least one phase parameter may beapplied to at least one row in the beamforming matrix. For example, thephase matrix and magnitude matrix may be multiplied as shown in equation(11) or equation (28). Further, the beamforming matrix may be used inbeamforming as shown in equation (28).

FIG. 4 shows a system diagram in accordance with various embodiments ofthe present invention. Electronic system 400 includes antennas 410,physical layer (PHY) 430, media access control (MAC) layer 440, Ethernetinterface 450, processor 460, and memory 470. In some embodiments,electronic system 400 may be a station capable of factoring beamformingmatrices and quantizing parameters as described above with reference tothe previous figures. In other embodiments, electronic system may be astation that receives quantized parameters, and performs beamforming ina MIMO system. For example, electronic system 400 may be utilized in awireless network as station 102 or station 104 (FIG. 1). Also forexample, electronic system 400 may be a station capable of performingthe calculations shown in any of the equations (1)-(28), above.

In some embodiments, electronic system 400 may represent a system thatincludes an access point or mobile station as well as other circuits.For example, in some embodiments, electronic system 400 may be acomputer, such as a personal computer, a workstation, or the like, thatincludes an access point or mobile station as a peripheral or as anintegrated unit. Further, electronic system 400 may include a series ofaccess points that are coupled together in a network.

In operation, system 400 sends and receives signals using antennas 410,and the signals are processed by the various elements shown in FIG. 4.Antennas 410 may be an antenna array or any type of antenna structurethat supports MIMO processing. System 400 may operate in partialcompliance with, or in complete compliance with, a wireless networkstandard such as an 802.11 standard.

Physical layer (PHY) 430 is coupled to antennas 410 to interact with awireless network. PHY 430 may include circuitry to support thetransmission and reception of radio frequency (RF) signals. For example,in some embodiments, PHY 430 includes an RF receiver to receive signalsand perform “front end” processing such as low noise amplification(LNA), filtering, frequency conversion or the like. Further, in someembodiments, PHY 430 includes transform mechanisms and beamformingcircuitry to support MIMO signal processing. Also for example, in someembodiments, PHY 430 includes circuits to support frequencyup-conversion, and an RF transmitter.

Media access control (MAC) layer 440 may be any suitable media accesscontrol layer implementation. For example, MAC 440 may be implemented insoftware, or hardware or any combination thereof. In some embodiments, aportion of MAC 440 may be implemented in hardware, and a portion may beimplemented in software that is executed by processor 460. Further, MAC440 may include a processor separate from processor 460.

In operation, processor 460 reads instructions and data from memory 470and performs actions in response thereto. For example, processor 460 mayaccess instructions from memory 470 and perform method embodiments ofthe present invention, such as method 200 (FIG. 2) or method 300 (FIG.3) or methods described with reference to other figures. Processor 460represents any type of processor, including but not limited to, amicroprocessor, a digital signal processor, a microcontroller, or thelike.

Memory 470 represents an article that includes a machine readablemedium. For example, memory 470 represents a random access memory (RAM),dynamic random access memory (DRAM), static random access memory (SRAM),read only memory (ROM), flash memory, or any other type of article thatincludes a medium readable by processor 460. Memory 470 may storeinstructions for performing the execution of the various methodembodiments of the present invention. Memory 470 may also storebeamforming matrices or beamforming vectors.

Although the various elements of system 400 are shown separate in FIG.4, embodiments exist that combine the circuitry of processor 460, memory470, Ethernet interface 450, and MAC 440 in a single integrated circuit.For example, memory 470 may be an internal memory within processor 460or may be a microprogram control store within processor 460. In someembodiments, the various elements of system 400 may be separatelypackaged and mounted on a common circuit board. In other embodiments,the various elements are separate integrated circuit dice packagedtogether, such as in a multi-chip module, and in still furtherembodiments, various elements are on the same integrated circuit die.

Ethernet interface 450 may provide communications between electronicsystem 400 and other systems. For example, in some embodiments,electronic system 400 may be an access point that utilizes Ethernetinterface 450 to communicate with a wired network or to communicate withother access points. Some embodiments of the present invention do notinclude Ethernet interface 450. For example, in some embodiments,electronic system 400 may be a network interface card (NIC) thatcommunicates with a computer or network using a bus or other type ofport.

Although the present invention has been described in conjunction withcertain embodiments, it is to be understood that modifications andvariations may be resorted to without departing from the scope of theinvention as those skilled in the art readily understand. Suchmodifications and variations are considered to be within the scope ofthe invention and the appended claims.

1. A method performed by a wireless station in a closed loopmultiple-input-multiple-output (MIMO) wireless network, the methodcomprising: receiving a signal transmitted by N antennas; determining abeamforming matrix from the signal; factoring phase information out ofcolumns of the beamforming matrix; representing the beamforming matrixwithout the phase information using N²−N angles; and transmitting theangles from the wireless station.
 2. The method of claim 1 furthercomprising quantizing the N²−N angles prior to transmitting.
 3. Acomputer-readable medium having instructions stored thereon that whenexecuted result in a wireless station in a closed loopmultiple-input-multiple-output (MIMO) wireless network performing:determining a beamforming matrix from a signal received from N transmitantennas; factoring phase information out of columns of the beamformingmatrix; representing the beamforming matrix without the phaseinformation using N²−N angles; and transmitting the angles from thewireless station.
 4. The computer-readable medium of claim 3 wherein theinstructions further result in the wireless station quantizing the N²−Nangles prior to transmitting.
 5. An electronic system comprising: Nantennas; a processor coupled to the N antennas; and an article having amachine-readable medium encoded with instructions that when accessedresult in the processor estimating channel state information from areceived signal, determining an N×N beamforming matrix from the channelstate information, factoring a phase angle out of each column of the N×Nbeamforming matrix, and representing the beamforming matrix without thephase angles using N²−N angles.
 6. The electronic system of claim 5wherein the instructions further result in the processor quantizing theN²−N angles.
 7. The electronic system of claim 6 wherein theinstructions further result in the electronic system transmitting theN²−N angles.